Soal Olimpiade Matematika Tingkat Internasional

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Soal Olimpiade Matematika Tingkat Internasional

Version : English

First day

Tokyo, July 13 2003

Problem 1. Let A be a subset of the set S = {1, 2, ...., 1000000} containing exactly 101 elements. Prove that there exist numbers t₁, t₂, ..., t₁₀₀ in S such that the sets :

A_j = {x + t_j | x ∈ A} for j = 1, 2, ..., 100

are pairwise disjoint.

Problem 2. Determine all pairs of positive integers (a, b) such that

is a positif integer.

Problem 3. A convex hexagon is given in which any two opposite sides have the following properti : the distance between their midpoints is √3/2 times the sum of their lenghts. Prove that all the angels of the hexagon are equal.

(A convex hexagon ABCDEF has three pairs of opposite sides : AB and DE, BC and EF, CD and FA.)

Second day

Tokyo, July 14, 2003

Problem 4. Let ABCD be a cyclic quardrilaterial. Let P, Q and R be the feet of the perpendiculars from D to lines BC, CA and AB respectively. Show that PQ = QR if and only if the besectors of ∠ABC and ∠ADC meet on AC.

Problem 5. Let n be a positive integer and x₁, x₂, ..., x_n be real numbers with x₁ < x₂ < .... x_n.

(a) Prove that :

(b) Show that equality holds if and only if ₁, x₂, ..., x_n is an arithmatic sequence.

Problem 6. Let p be a prime number. Prove that there exist a prime number q such that for every integer n, the number n^p-p is not divisible by q.

Sekian artikel kali ini. Mohon maaf apabila ada salah-salah kata.

Akhir kata wassalamualaikum wr. wb.

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